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Time-changed generalized fractional Skellam process

Mostafizar Khandakar, Bratati Pal, Palaniappan Vellaisamy

TL;DR

This work addresses non-Markovian count processes by time-changing the generalized Skellam framework. It introduces two variants, the time-changed generalized fractional Skellam process-I (TCGFSP-I) and the time-changed generalized fractional Skellam process-II (TCGFSP-II), constructed via independent Lévy subordinators and their inverses, respectively, and derives comprehensive distributional and functional-characterization tools for them. The authors obtain explicit pmf/pgf/mgf representations, factorial moments, means, variances, covariances, and governing differential equations, and prove overdispersion and long-range dependence under appropriate subordinator assumptions; they also establish a law of iterated logarithm for TCGFSP-I and explore several concrete subordinators (gamma, tempered stable, inverse Gaussian). The results yield versatile, analytically tractable models for non-Markovian, long-memory counting phenomena with potential applications in finance, hydrology, internet traffic, and related fields. By linking subordination, fractional calculus, and Skellam-type structures, the paper provides a unified framework for analyzing time-changed Skellam processes with rich dependence and dispersion properties.

Abstract

In this paper, we introduce and study two time-changed variants of the generalized fractional Skellam process. These are obtained by time-changing the generalized fractional Skellam process with an independent Lévy subordinator with finite moments of any order and its inverse, respectively. We call the introduced processes the time-changed generalized fractional Skellam process-I (TCGFSP-I) and the time-changed generalized fractional Skellam process-II (TCGFSP-II), respectively. The probability generating function, moment generating function, moments, factorial moments, variance, covariance, {\it etc.}, are derived for the TCGFSP-I. We obtain a variant of the law of the iterated logarithm for it and establish its long-range dependence property. Several special cases of the TCGFSP-I are considered, and the associated system of governing differential equations is obtained. Later, some distributional properties and particular cases are discussed for the TCGFSP-II.

Time-changed generalized fractional Skellam process

TL;DR

This work addresses non-Markovian count processes by time-changing the generalized Skellam framework. It introduces two variants, the time-changed generalized fractional Skellam process-I (TCGFSP-I) and the time-changed generalized fractional Skellam process-II (TCGFSP-II), constructed via independent Lévy subordinators and their inverses, respectively, and derives comprehensive distributional and functional-characterization tools for them. The authors obtain explicit pmf/pgf/mgf representations, factorial moments, means, variances, covariances, and governing differential equations, and prove overdispersion and long-range dependence under appropriate subordinator assumptions; they also establish a law of iterated logarithm for TCGFSP-I and explore several concrete subordinators (gamma, tempered stable, inverse Gaussian). The results yield versatile, analytically tractable models for non-Markovian, long-memory counting phenomena with potential applications in finance, hydrology, internet traffic, and related fields. By linking subordination, fractional calculus, and Skellam-type structures, the paper provides a unified framework for analyzing time-changed Skellam processes with rich dependence and dispersion properties.

Abstract

In this paper, we introduce and study two time-changed variants of the generalized fractional Skellam process. These are obtained by time-changing the generalized fractional Skellam process with an independent Lévy subordinator with finite moments of any order and its inverse, respectively. We call the introduced processes the time-changed generalized fractional Skellam process-I (TCGFSP-I) and the time-changed generalized fractional Skellam process-II (TCGFSP-II), respectively. The probability generating function, moment generating function, moments, factorial moments, variance, covariance, {\it etc.}, are derived for the TCGFSP-I. We obtain a variant of the law of the iterated logarithm for it and establish its long-range dependence property. Several special cases of the TCGFSP-I are considered, and the associated system of governing differential equations is obtained. Later, some distributional properties and particular cases are discussed for the TCGFSP-II.

Paper Structure

This paper contains 15 sections, 9 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mittag-Leffler function
  4. Subordinator and its inverse
  5. Stable and inverse stable subordinator
  6. The GSP and its time-changed variant
  7. The LRD and SRD properties
  8. Time-changed generalized fractional Skellam process
  9. Some special cases of the TCGFSP-I
  10. GFSP time-changed by gamma subordinator
  11. GSP time-changed by tempered stable subordinator
  12. GSP time-changed by inverse Gaussian subordinator
  13. GFSP time-changed by inverse subordinator
  14. GSP time-changed by the inverse TSS
  15. GSP time-changed by the first passage time of IGS

Key Result

Proposition 3.1

The $r$th factorial moment $\Psi_f^\alpha(r, t) = \mathbb{E}(\mathcal{Z}^\alpha_f(t) (\mathcal{Z}^\alpha_f(t) - 1) \cdots (\mathcal{Z}^\alpha_f(t) - r + 1))$, $r \geq 1$ of TCGFSP-I is given by where $j^{(m_l)} = j(j+1) \cdots (j+m_l-1)$ denotes the rising factorial and $(J)_{m_l} = j(j-1) \cdots (j-m_l+1)$ denotes the falling factorial.

Theorems & Definitions (25)

  • proof
  • proof
  • Proposition 3.1
  • proof
  • proof
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  • Proposition 3.2
  • proof
  • proof
  • ...and 15 more